Problem: Simplify and expand the following expression: $ \dfrac{4}{x - 5}+ \dfrac{5}{2x - 10}+ \dfrac{1}{x^2 - 10x + 25} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the second term: $ \dfrac{5}{2x - 10} = \dfrac{5}{2(x - 5)}$ We can factor the quadratic in the third term: $ \dfrac{1}{x^2 - 10x + 25} = \dfrac{1}{(x - 5)(x - 5)}$ Now we have: $ \dfrac{4}{x - 5}+ \dfrac{5}{2(x - 5)}+ \dfrac{1}{(x - 5)(x - 5)} $ The least common multiple of the denominators is: $ (x - 5)(x - 5)$ In order to get the first term over $(x - 5)(x - 5)$ , multiply by $\dfrac{2(x - 5)}{2(x - 5)}$ $ \dfrac{4}{x - 5} \times \dfrac{2(x - 5)}{2(x - 5)} = \dfrac{8(x - 5)}{(x - 5)(x - 5)} $ In order to get the second term over $(x - 5)(x - 5)$ , multiply by $\dfrac{x - 5}{x - 5}$ $ \dfrac{5}{2(x - 5)} \times \dfrac{x - 5}{x - 5} = \dfrac{5(x - 5)}{(x - 5)(x - 5)} $ In order to get the third term over $(x - 5)(x - 5)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{1}{(x - 5)(x - 5)} \times \dfrac{2}{2} = \dfrac{2}{(x - 5)(x - 5)} $ Now we have: $ \dfrac{8(x - 5)}{(x - 5)(x - 5)} + \dfrac{5(x - 5)}{(x - 5)(x - 5)} + \dfrac{2}{(x - 5)(x - 5)} $ $ = \dfrac{ 8(x - 5) + 5(x - 5) + 2} {(x - 5)(x - 5)} $ Expand: $ = \dfrac{8x - 40 + 5x - 25 + 2}{2x^2 - 20x + 50} $ $ = \dfrac{13x - 63}{2x^2 - 20x + 50}$